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SANDIA REPORT SAND96-8212 UC-1409 Unlimited Release

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n

Combustion of Porous Energetic Materials in the Merged-Flame Regime

Stephen B. Margolis, Forman A. Williams, Alexander M. Telengator

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a

ore, California 94551

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SAND96-8212 0 UC-1409

Unlimited Release

Printed February 1996

COMBUSTION O F POROUS ENERGETIC MATERIALS

IN THE MERGED-FLAME REGIME

STEPHEN B. MARGOLIS

Combustion Research Facility

Sandia National Laboratories

Livermore, California 94551 USA

and

FORMAN A. WILLIAMS and ALEXANDER M. TELENGATOR

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego

La Jolla, California 92093 USA

Abstract

The structure and burning rate of an unconfined deflagration propagating through a porous energetic material is analyzed in the limit of merged condensed and gas-phase reaction zones. A

global two-step reaction mechanism, applicable to certain types of degraded nitramine propellants

and consisting of sequential condensed and gaseous steps, is postulated. Taking into account

important effects due to multiphase flow and exploiting the limit of large activation energies, a theoretical analysis based on activation-energy asymptotics leads to explicit formulas for the

deflagration velocity in a specifically identified regime that is consistent with the merged-flame

assumption, The results clearly indicate the influences of two-phase flow and the multiphase,

multi-step chemistry on the deflagration structure and the burning rate, and define conditions that

support the intrusion of the primary gas flame into the two-phase condensed decomposition region at the propellant surface.

COMBUSTION O F POROUS ENERGETIC MATERIALS

I N THE MERGED-FLAME REGIME

Introduction

There is increasing interest in the combustion behavior of systems characterized by significant

two-phase flow. Examples include filtration combustion, smoldering, and the deflagration of porous

energetic materials, where the latter, the focus of the present work, is of interest in propulsion and

pyrotechnics. In such problems, the porous nature of the material arises from a certain degree of

metastability which, after either a prolonged existence and/or exposure to an abnormal thermal

environment, leaves the material in a chemically degraded, porous state. As a result, two-phase-

flow effects associated with different velocities and properties of the condensed and gaseous species

have a pronounced effect on the structure and propagation velocity of the combustion wave.

Although relatively complete formulations have been proposed for analyzing combustion phe-

nomena involving multiphase flow [l], they are difficult to analyze because of the wide range of

physical phenomena associated with such systems and the highly nonlinear nature of the problem.

Accordingly, early two-phase work in this area tended to alleviate some of the difficulties by treat-

ing the two-phase medium as a single phase with suitably averaged properties [2,3]. Unfortunately,

such models effectively require the velocity of each phase to be the same, precluding any analysis

of two-phase-flow effects. More recently, however, it has proven possible to analyze deflagration

models for porous energetic materials that explicitly involve multiphase flow [4-91. These studies

have largely been applicable to nitramine propellants, such as HMX and, in some cases, RDX,

that are characterized by a liquid melt region in which extensive bubbling in an exothermic foam

layer occurs. In some cases [4-61, two-phase-flow effects were confined to this layer, while in others

[7-91, the solid material was assumed to be porous, with two-phase flow occurring throughout the

preheat region. In order to focus on the effects of multiphase flow, chemistry was generally confined

to a single-step overall reaction R(c) + P(g) representing direct conversion of condensed (melted)

propellant to gaseous products. A generalization [5] in which a separate (primary) gas flame follows

the initial multiphase decomposition region is given by R(c) + P(g) , R(c) * R(g), R(g) --+ P(g) ,

where R(g) is a gaseous reactant. This scheme was applied to a nonporous problem to determine

the structure and propagation velocity of a steady, planar nitramine deflagration, while stability

results for these models have thus far been confined to single-step mechanisms [6,8,9].

The present work analyzes the limiting case in which the primary gas flame intrudes upon the

multiphase decomposition region, a tendency that is often observed experimentally as the pressure

5

increases. Thus, the single-step analysis [7] is extended by incorporating both condensed- and gas-

phase reactions in the thin multiphase reaction region. In particular, a global sequential reaction

mechanism is assumed, consisting of an overall condensed-phase reaction that produces gas-phase

intermediates, and a gas-phase reaction that converts these intermediates to final products. While

this mechanism is still an extreme approximation to actual nitramine chemistry [10,11], it enables

us to fully incorporate two-phase flow into the analysis, and to assess its effect on the structure

and propagation speed of the deflagration. The merged-flame analysis presented here is thus

a multiphase-flow analog to single-phase studies of propagating combustion waves that are also

governed by sequential reactions occurring in a single thin reaction zone [12-151.

The Mathematical Model

We consider an unconfined, steady, planar deflagration, propagating from right to left into a

degraded (porous) energetic solid. Melting of the solid occurs at a moving location 5 = Zm(t"> where

the solid temperature equals its melting point Tm. Subsequent to melting, gas-phase intermediates

are assumed to be produced directly by liquid-phase reactions, and these, in turn, react to form

final combustion products according to the mechanism R(c) 4 I(g), I(g) -+ P(g), where R(c), I(g)

and P(g) denote the condensed (melted) reactant material, the intermediate gas-phase species, and

the final gas-phase products. The pores within the solid are assumed to be filled with a mixture

of I ( g ) and P(g) , with the mass-fraction ratio 4 of the two specified far upstream. The present

analysis considers a merged regime in which both reactions occur within a single reaction zone, and

thus, the deflagration wave consists of a solid/gas preheat region, the melting surface, a liquid/gas

preheat region, a thin reaction zone in which all reactive species are converted to gaseous products,

and the burned region, The latter typically corresponds to a dark zone that separates the primary

flame from a secondary gas flame that has little effect on the burning rate.

A model describing a multiphase deflagration was derived previously for a single-step reac-

tion mechanism R(c) + P(g) [7]. For two gas-phase species ( I and P ) , an additional species

conservation equation is thus required. For simplicity, we consider the single-temperature limit

in which the rates of interphase heat transfer are large so that all phases have the same local

temperature, but since a complete derivation of the full two-temperature model is given elsewhere

[7,16], we provide only a brief explanation of the origin of each equation. Thus, we introduce, in

terms of dimensional quantities (denoted by tildes) defined in the nomenclature, the nondimen-

sional variables 5 = j j s & ~ ~ / x s , t = jjsZsU2t/X,, Ts,r,g = ? s , r , g / ? ! , ul,g = iir,g/o, and pg = &/pi, where 0 = -&,/df is the (unknown) propagation speed of the melting front, and constant heat

- - -

6

capacities and thermal conductivities have been assumed. We also define the parameters

r = pL/b3, F = p ; / p s , I = Xl/Xs , i = X,/Xs , b = z 1 / z 3 , i = zg/Es, ys = 73/c"3Tu , w = W I / W p , Le = X,/p,bE, , Q1,, = Ql,,/EsTu, N1,, - - .?31$g/PTb ,

Ai = X s & e - N 1 / ~ 3 E 3 ~ 2 , Ag = X , $ ( $ ) " e - N g / f i ~ E s ~ 2 ,

where n is the reaction order of the gas-phase reaction and Le is the gas-phase Lewis number.

Here, x,, pgB and hence Le are assumed constant, r and F are density ratios (liquid-to-solid

and upstream gas-to-solid), I and are thermal conductivity ratios, b and 6 are heat-capacity

ratios, w is the ratio of molecular weights of the two gas-phase species, /̂s is a heat-of-melting

parameter (negative when melting is endothermic), Q L , ~ are heat-release parameters associated

with the condensed and gas-phase reactions, Nl,, are the activation energies, and AI,, are rate

coefficients, or Damkohler numbers. Since h,/Al = + ( ~ , / 8 s ) ( p ~ ) n - 1 e ~ ~ - ~ g , we may regard AI as

the burning-rate eigenvalue.

For a steadily propagating deflagration, it is convenient to transform to the moving coordinate

t = z + t whose origin is defined to be zm(t). In the solid/gas region, the volume fraction o of

gas is assumed constant (a = as), the solid phase is assumed to have constant density and zero

velocity with respect to the laboratory frame of reference, and gas-phase continuity is given by

representing overall continuity for the gas phase and mass conservation for the mass fraction Y of

the gas-phase intermediates. In the liquid/gas region > 0, overall continuity, continuity of the

liquid phase, and continuity of the intermediate gas-phase species are given as

d d Y Ns?(l-Tb/Tg) d - [+op,Y(u, + 1) +r(l - a ) ( u ~ + l)] = @/&Le)- [a-] - A,(ap,Y)"e dt d e d t

where Eq. (5) has been used to eliminate the condensed-phase reaction-rate term that would have

otherwise appeared in Eq. (6). Finally, overall energy conservation is given by

d - [r(1- Q)(uL + l ) (Qi + Qg + bT) + Papg(ug + l)(QgY + h)] d dT d Y

d 5 a = -{ [Z(l - a) + h] - + ( Q g i / 6 L e ) a z } , [ > 0 ,

7

where Eqs. ( 5 ) and (6) have been used to eliminate the reaction-rate terms in the latter.

The above system of equations is closed by adding an equation of state (assumed to be that

of an ideal gas) and an expression for the liquid velocity zcl. An approximate analysis of gas-phase

momentum conservation implies, for an unconfined, small Mach-number deflagration, that pressure

is constant, thereby allowing the former to be expressed as pgT [Y + ~ ( 1 - Y)] = 4 + w(1- 4) E

4, where 4 is defined below. An analysis of condensed-phase momentum and the assumption

of zero velocity for the solid phase, on the other hand, leads to the kinematic approximation

u1 = (1 - r ) / r in the limit of small viscous and surface-tension-gradient forces [4,7]. The problem

is then completely defined by imposing the boundary conditions a = as for 5 0, ug --+ 0, Y -+ 4

(0 5 5 l), T --$ 1 as e + -co and a + 1, Y -+ 0. T + T b as e + +m, and the melting-surface

conditions, which are Continuity of T = T,, ug, Y and dY/dt at E = 0 and the jump condition

-

[2(1 - as) + fas] - - (1 - as + f a s ) q d T 1 d t +o+ d t c=o- = (1 - as) [-3$ + (b - l)Tm] . (9)

Here, the final burned temperature Tb is to be determined, as is the burning-rate eigenvalue AI,

which determines the propagation velocity of the deflagration according to Eq. (2).

Expressions for Tb and the burned gas velocity u: ugIc=oo$ are obtained from the above

model as follows. From Eqs. (3), (4) and the boundary conditions, we have the first integrals

(10)

(11)

where p i = $/wTb is the burned gas density. Thus, evaluating Eq. (11) at = 0 using the first of

dY Y - 4 = @/&e) - d t ' P g b g + 1) = 1 9

(1 - a) + Fapg(ug + 1) = Fpg(ug + 1) ,

t < 0 9

t > 0 , b b

Eqs. (10) and continuity across the melting surface, we obtain

Ubg = [ 1 -I- as (F - I)] - 1 = [1+ as (e - l)] wT~/&$ - 1 a

Using these results, first integrals of the overall energy equations (7) and (8) are given by

dT (1 - as + FLs)(T - 1) = (1 - as + fas)-

ds' e < o

dT [ b ( l - a) -k ~ ( C U - as + aSP)]T + (CY - as + asf)QgY = [Z(1- a) + iOr] -

dt dY + ( Q g t / 6 L e ) a z - (1 - a)(&i + Qg) + 6(1- as + as'F)Tb 6 > 0,

Thus, subtracting Eq. (13) evaluated at t = 0- from Eq. (14) evaluated at .$ = O+ and using the

melting-surface condition (9), we obtain

8

This result, which can be derived from a more general two-temperature model [16], is independent

of the particular form of the gas-phase equation of state. In the limit Q9 -+ 0, Eq. (15) collapses to

the result obtained for the single-step model [7]. Equations (lo), (13) and (14)) being first integrals

of Eqs. (3), (7) and (8), now take the place of the latter in our model.

Equations (12) and (15) imply that there are significant variations in Tb and u: with the

upstream gas-to-solid density ratio f , which in turn is proportional to the pressure fli according

to P &/& = fi&/jjsp!f'u$. This important effect arises from the thermal expansion of the

gas, the two-phase nature of the flow in the solid/gas and liquid/gas regions, and the fact that, for

nonzero os, some of the heat released by combustion must be used to help raise the temperature

of the gas-phase species within the porous solid from unity to Ti [7]. Consequently, both T b and

u: are typically decreasing functions of ?. An additional effect, revealed by the two-step reaction

analysis, is that T b does not depend just on the total heat release Q1 + Q9 = Q associated with

the complete conversion of the energetic solid to final gas products, but also on the heat release

Q9 specifically associated with the gas-phase reaction. This, too, is a two-phase-flow effect that

arises from the fact that reactive intermediate species exist within the voids in the porous solid,

and the heat released by these intermediates affects the final burned temperature, which, for a

given total heat release Q, increases as the fractional heat release associated with the gas-phase

reaction increases.

The Asymptotic Limit and the Outer Solution

Further analytical development leading to the determination of A1 requires an analysis of

the reactive liquid/gas region 6 > 0. Equations (5), (6) and (14) constitute three equations

for Y, T and Q in this region, with u9 then determined from Eq. (11) and the equation of

state, and the eigenvalue hl determined by the boundary conditions. In order to handle the

Arrhenius nonlinearities in Eqs. (5) and (6), we exploit the largeness of the activation energies

Ng,r and consider the formal asymptotic limit Ng,l >> 1 such that Ng/Nl = v N 0(1) and P = (1 -Tcl)Nl >> 1, where f l is the Zel'dovich number. This ordering of the activation energies, along

with a corresponding order relation for the ratio Ag/Al to be introduced shortly, helps to insure

that both the condensed and gas-phase reactions are active in a single thin reaction zone, since

departures from these orderings can result in separated reaction zones [17].

In the limit f l 4 03, the Arrhenius terms are exponentially small unless T is within O(l/fl)

of Tb. Consequently, all chemical activity is concentrated in a zone whose thickness is O(l/p).

On the scale of the (outer) coordinate e, this thin region is a sheet whose location is denoted

by & = xT - X m , where xcT > xm. Hence, the liquid/gas region is comprised of a preheat zone

9

(0 < e < er) where chemical activity is exponentially small, the thin reaction zone where the two

chemical reactions are active and go to completion, and a burned region 6 > &. Denoting the

outer solutions on either side of the reaction zone by a zero superscript, we conclude from Eq. ( 5 ) ,

(10) - (12) and the equation of state,

where the latter is valid for all e. Thus, there is a jump in a', and hence in u:, across the reaction

zone. Similarly, in obtaining the complete outer solution for Y and T , it is necessary to connect

the solutions on either side of the reaction zone by deriving appropriate jump conditions across

c = This will entail an analysis of the inner reaction-zone structure, whereupon an asymptotic

matching of the inner and outer solutions will yield these jump conditions and an expression for

hi. In connection with this procedure, it is convenient, and physically appealing, to attempt a

representation of the reaction-rate terms in Eqs. (5) and (6) as delta-function distributions with

respect to the outer spatial variable [12,13]. Using the results (16)) the system of equations for

the outer variables Y o and To thus become

(17) dY' dolo

dE t < 0; (1/~)- = R S ( F - [r) t > 0 2 ?(YO - 4) = ( i / i L e ) x

d _rG (CY'%) - P g 6 ( [ - & . - - H ) , e > O , (18) - [(CY' - as + +cY,)Y' - CY'] = ( l / iLe ) 4 d5

A dTo (1 - CY, + ?iff,) (TO - 1) = (1 - CY, + la,) - dl )

PO

where

A sketch of the outer solution is shown in Fig. 1. Since the interval & < < < H lies within the

merged reaction zone, only that portion of Eqs. (21) and (22) for < & and ,$ > tT + H actually

represents the outer solution. Consequently, there is an O ( H ) jump in Yo and To across this zone

[from 6 = e; to 6 = (& + H)+]. Alternatively, for small H , an expansion of S(S - ST - H ) in Eq.

(18) about H = 0 introduces the derivative S'(6 - &) [13], which implies discontinuities in Yo and

To at C = &-e These discontinuities are determined by H , which, like 111 , is an eigenvalue. Both

are calculated by matching the outer solution to that of the inner reaction-zone problem.

Reaction-Zone Solutions

To analyze the chemical boundary layer at &., we introduce a stretched inner variable =

@(.$-&) and a normalized temperature 0 = (T- 1 ) / ( T b - 1). We then seek solutions in this region

as a - a0 +p- la' + * * , u g - uo +p-' u1 + ... , Y - p-'yl +p-292 + - . - , o - 1 +p-lel + -. . , AI - O(Ao +@-'AI +. - e ) and H - P-'hl +pw2hz +. a , where the ui are calculated in terms of the

ai, yi and 8i from Eq. (16), which is valid in the reaction zone. We also order the rate-coefficient

ratio Ag/Ar = F ( i j t ) n - ' ( & / ~ l ) e('--y)Ni E p"X, where X is an 0(1) parameter. This scaling, along

with the previous ordering of the activation energies, partly defines one regime that is consistent

with a merged reaction-zone structure.

Substituting these expansions into Eqs. (5), (6) and (14), the leading-order inner variables

00, y1 and 81 are governed by

Solutions to these equations as 77

t 1 (eT + H)+, leading to the matching conditions

f m must match with the outer solution as < 1' .$; and as

b( 1 - a,) + &as 6 (28)

Z(1 - a,) + ias E1 = [ ( T b - B)/(Tb - 1)J , E2 = -[(T* - Bl)/(Tb - 1)]1(1 -as ++as).

1

11

Solution of the complete inner problem given by Eqs. (27) - (31) will only be possible for certain

values of hl and Ao, the leading-order coefficients in the expansions of H and Ar.

Employing a0 as the independent variable, Eqs. (25) and (26) take the form

d0i dYl da0 da0

[Z + (i - Z ) ~ O ] eel - + [iQg/6Le(Tb - l)]aoeel - = [(b - 6 ) T b + QI + Qg]/(Tb - l)rAo, (29)

ao(1 -a0)ee1*] = -1 + (X/r)(aOyl~//WTb)ne(Y-')el/( l - ao) e

da0

A closed-form solution to this system is not readily apparent, and thus further analytical develop-

ment is restricted to a perturbation analysis of Eqs. (29) and (30) in the limit that Qg is small

relative to Ql. Since this implies that most of the heat release occurs in the first stage of the two-

step reaction process, at least some of the initial exothermic gas-phase decomposition reactions

should be lumped with the overall reaction (la), regarding the resulting decomposition products

as the gas-phase intermediates I ( g ) . Thus, we formally introduce a small bookkeeping parameter E ,

where O(j3-') << E << O ( l ) , and write Qg = to:, where Qt - O(1). We now seek solutions to the

leading-order inner problem in the form a0 - a: + ea: +. - , y1 - 9: +cy: + ,61 - 8: + €0; + - , A0 - A: + + 0 , and hl - hy + rh{ + 9 - m .

Substituting these latest expansions into Eqs. (24), (29) and (30), a closed subsystem

is obtained, subject to a: -+ 1, 0: -+ 0 as q 4 +m, and a: -+ a,, 6: - Etq as q --f -m, where

E! is given by E1 in Eq. (28) with T' replaced by its leading-order approximation B1. The first

of Eqs. (31) is readily integrated from a: = a, (at q = -00) to any a! I 1 (q 5 +m) to give

eeY(4) = [ (b - 6)B1/(B1 - l)rhg] [' a/[Z + (i - l ) ~ ]

{ [(b - 6 ) @ + QI] / (BI - l)r(i - Z)} In {i/[Z + (i - Z)Q,]}, 1 # i 1 = i .

Evaluating Eq. (32) at a: = 1 (at which 0: = 0) thus determines A: as

{ (1 - as>[(b - b)B1+ Qr]/(B1 - 1)rh A: =

Substituting this result into Eq. (32) for arbitrary QO, we thus obtain

(33)

The determination of a;(q), and hence @(q) , then follows directly from the second of Eqs. (35).

For example, when i = E (equal gas and liquid thermal conductivities), we obtain

12

where the matching condition at 7 = -cm has been used to evaluate the constant of integration.

From Eq. (33), the leading-order expression for A1 is independent of the effects of the second

reaction, which has been assumed to have a relatively small thermal effect. Consequently, the first

effects of the two-step mechanism on the burning rate appear at O(E) , which requires the calculation

of A;. We proceed by first calculating yy, which is determined from the leading-order version of

Eq. (30). For simplicity, we consider the parameter regime as = a:€, I^ = E + d1 and v = 1 + ~ d , corresponding to O(E) porosities, O(E) differences in the conductivities of the condensed and gaseous

phases, and O(@) differences in the activation energies of the two reaction steps. Also, we consider

only a first-order gas-phase reaction (n = l) , and assume that (A3/rwAgTt)(bLe/rZ) = A0 + CAI,

where Tf = (Ql + 1 + ys)/b is the leading-order approximation to Tb with respect to E . The

parameter group (A3/rwAtTf)( 6LelrZ) is a gas-to-liquid ratio of diffusion-weighted reaction rates,

referred to as consumption rates [12,13], which represent relative rates of depletion, taking into

account both chemical reaction and, for the gas phase, species diffusion. The fact that larger Lewis

numbers are associated with higher rates of depletion of the gaseous reactant stems from the higher

concentration of this species in the reaction zone that results from a smaller mass diffusivity.

In the parameter regime just outlined, 6: and A! simplify to

or 7 = (l/rAg) ln[a8/(1 - a:)]. Hence, the leading-order version of Eq. (30) for 2: is given by

subject to 3: --t 0 as cy: 1. An appropriate matching condition as a: --t 0, however, cannot

be obtained directly from Eq. (28) because that equation was derived under the assumption that

as # 0, whereas to leading order in E , as is zero. Indeed, at this level of approximation, the outer

variable Yo has no meaning for < 0, since there is essentially no gas in this region. To derive

proper matching conditions on yf and, for later use, yl, we consider the local mass fraction 2

of the intermediate gas-phase species with respect to the total mass of all species, gaseous and

condensed [16]. Thus, 2 = Pap,Y/[Pa/p,+r(l -a)] = P&Y/{P&r+r(l -a)[Y + w ( l -Y)]T} is

unambiguous as a -+ 0, where it must vanish. From the outer solution written in terms of 7, the

required behavior of 2 as 7 --f -m is 2 N Ea-'(P3at~Le/rw~fE)(-Pdrl+hy). Hence, substituting

the inner expansions into the definition of 2 and imposing this asymptotic behavior implies

where we have written these conditions in terms of a:, using the fact that a: 4 af as ai 4 0.

Formal solutions to Eq. (37) can be expressed in terms of hypergeometric functions, but since

we desire explicit representations for use in the next-order problem, we focus on such solutions

that may be obtained for certain values of XO. In particular, solutions for Xo = 1 [IS] and XO = 4

that satisfy the matching conditions are given by

We observe that although agyy 4 0 as at 4 0, 9: itself is unbounded, exhibiting the behavior

Profiles of the leading-order inner variables as functions of are shown in Fig. 2.

The reaction-zone problem at the next order in E determines ai, y:, 6; and Ai , as well as hy,

which is still unknown. As before, the problem for CY: and 6; decouples and is given by

where

and Tb - Tt +ET; +. * , with Tt = (Qt + 1 +?$)/6 and Ti = Qi/b - ?(Tt - 1)~:. We observe that

Tb9 and hence C1 and E!, all depend on ai, reflecting, to this order of approximation, a linearly

decreasing dependence of the burned temperature on the porosity of the solid.

Tkansforming to a: as the independent variable, Eqs. (41) and (42) become

(45)

subject to Eq. (43) expressed in terms of a:. Rewriting Eq. (45) as (d/dag)[ai/(l - a:)] = (0; + AA/A:)/(l - a:) and substituting this result into the derivative of Eq. (46), we obtain

14

where 9: is given by Eq. (39). Homogeneous solutions of Eq. (47) are (1-ai) In [az(l - +1

and 1 - ai, so that the general solution for 0; may be constructed by reduction of order and

substituted into Eq. (45) to determine a:. Thus, for example,

(49)

for A0 = 4, where c1, c2 and c3 are integration constants, and the polylogarithms Lin(a), n 2 2, are

defined recursively for all complex a by Li2(a) = - 1; &-' ln(1- &) d& = aj / j2 , Li,>2 (a) =

&-lLin-l(&) d& = E;, aj/j", where the series representation is convergent for la1 5 1 [HI.

For 0 5 cr 5 1, Li2(a) and Li3(a) are monotonic functions that range from Liz(0) = Li3(O) = 0 to

Liz(1) = 7r2/6 and Li3(l) = 1.20205690. Also, Liz(a) + Liz(1- a) = 7r2/6 - h a ln(1- a), which

was used to obtain Eqs. (48) and (49). Application of the matching conditions then determines

Ah, c1, c2(hy) and c3 as

A:=Qi(21-2.rr2)/6rZ(T:-l) + E,'/?- - At12/21, c2 = Qi{ [3-.rr2+6Li3(1)]/3rA; - h:}/Z(T:-l), CI = -Qi(9+x2)/3rZA:(T,0-1) + E;/rA: - f1/2Z, ~3 =af + Qi(2n2-3)/6~ZA8(T,0-1). (50)

An expression for hy may be determined by continuing with the perturbation analysis, but it

can be deduced directly from the second matching condition (38) which, for A0 = 4, gives

Since 1/cr; - exp(-rA:r)) as r ) + -00, Eq. (51) implies that yi grows exponentially as r) + -m

unless the right-hand side is identically zero. Since only algebraic growth of the inner solution

is compatible with an asymptotic matching with the outer solution, as indicated in Eq. (28), we

conclude from Eq. (51) that ?# = 1 + O(E) and h: = (-3/2)/(rA:). This required restriction of

? to values that are relatively close to l/# 1 1 corresponds to high upstream gas-phase densities,

15

or pressures, and may be interpreted as a compatibility condition, required for the existence of a

merged-flame solution, that accompanies our ordering of the activation energies and consumption

rates when gas-phase heat release is small. According to Eq. (16) for ug, it essentially limits

the twu-phase-flow effect to that associated with thermal expansion of the gas. Larger rates of

gas-phase transport relative to the condensed phase would cause the gas-phase reaction to occur

increasingly downstream of the condensed reaction, leading to a breakdown in the merged-flame

structure analyzed here, but it is anticipated that larger gas-phase consumption rates would allow

for larger gas-phase convective transport arising from smaller upstream gas densities. Profiles of

a: and St for A0 = 4 are shown in Fig. 2.

Discussion of the Burning Rate and Conclusions

The dimensional propagation speed 0, from the definition of AI given in Eq. (2)' is given by

0 2 - ( isAl /ps )PO [I +ET: (2 - T,O)/T,O (T,O - 1) + . -3 (A: + €A: + . - .)-le-Np(l-'T,'/T,"+...) (52) - ( XsAl / p 8 cs ) (Po/A~) , -N~eBsoT~/(T~- l )+' . ' { 1 + E[T: (2 - Tt)/T? (Tt - 1) - Ai/A:] + - e} ,

where Tb, which appears in the definitions of the nondimensional activation energy Nl and the

Zel'dovich number p , has been expanded according to the expansion given below Eq. (44), and we

have introduced the €-independent definitions Np = fil/&Op: and Po = (T; - l)N:/T'. Substi-

tuting the expressions obtained in the previous section for A! and A; and setting the bookkeeping

parameter E equal to unity (implying a: as, i' z i - I and Qi = Qg), we obtain the asymptotic

expression for the burning rate, in the specific merged-flame parameter regime considered here, as

for A0 = 4, with a similar expression for A0 = 1. The first effects of heat release associated with

the second step of the reaction model are determined by the terms proportional to Q9 in Eq. (53).

The dominant effects associated with gas-phase heat release are determined by the second

exponential factor in Eq. (53), which is exponentially large unless a,/&, x k' (T; - l)-'$. Val-

ues of a,/&, less (greater) than this critical value thus produce a significant increase (decrease)

in the burning rate over that of a nonporous material governed solely by the condensed reaction,

corresponding to whether or not the perturbation in the burned temperature, which arises from

nonzero porosity and the additional heat release associated with the gas-phase reaction, is posi-

tive or negative- Since Q9 is positive, the additional heat release associated with the gas-phase

16

reaction enhances the burning rate, but decreasing amounts of solid material that correspond to

increasing porosities lead to a lower overall heat release associated with the condensed-phase reac-

tion, resulting in a critical value of porosity for which these counteracting effects balance. Plots of

U = P(Qg, a s ) / ~ ( O , 0) are shown in Fig. 3.

Although the primary effect associated with nonzero porosity and a second gas-phase reaction

step is thus thermodynamic in nature, additional effects are revealed by those terms arising from

the correction A: to the leading-order burning-rate eigenvalue A:, which give rise to the last two

terms proportional to both Q9 and as and the term proportional to 1- I within the curly brackets

in Eq. (53). For example, it is readily seen that a value of the gas-phase thermal conductivity

greater (less) than that of the liquid phase tends to increase (decrease) the propagation speed,

since larger values of i - 1 allow for greater heat transport from the reaction zone back to the

preheat region, providing a type of “excess enthalpy” effect for the condensed phase portion of

the reaction. The result for the case in which XO = 1, corresponding to a smaller pre-exponential

reaction-rate coefficient for the gas-phase reaction, is identical to Eq. (53), except for the fact that

the numerator 21 - 2n2 A 1.26 is replaced by 7r2 - 6 A 3.87 in the second term proportional to

Qg in that equation. Thus, as expected, a smaller gas-phase consumption rate results in a smaller

overall burning rate when the corresponding rate for the condensed-phase reaction is unchanged.

In conclusion, the present analysis has sought to describe some of the effects associated with

the deflagration of porous energetic materials arising from two-phase-flow in the presence of a

multiphase sequential reaction mechanism. In contrast to previous work in which the condensed

and gas-phase reactions were spatially separated, a merged-flame parameter regime, in which both

reactions are operative and proceed to completion in a single thin reaction zone, was considered in

the present study. Although additional parameter constraints were required to support a merged-

flame structure, such a structure was calculated for the case of a high-pressure deflagration in

which the relative flow of gas with respect to the condensed material arises primarily from thermal

expansion of the former. This result is consistent with typical experiments involving the nitramine

propellants HMX and FtDX that show the tendency of the primary gas flame to move closer to the

propellant surface as the pressure increases. Further parametric studies are in progress and will be

reported in future publications.

Nomenclature

pre-exponential rate coefficient

mass diffusion coefficient

heat capacity

activation energy

heat release

gas constant

time

temperature variable

velocity variable

propagation speed

molecular weight

spatial coordinate

gas-phase mass-fraction variable

gas-phase volume fraction

heat of melting

thermal conductivity

density

gas-phase mass-fraction ratio

Subscripts, superscripts:

g, E , s gas, liquid, solid phase

b downstream burned value

rn melting-surface value

u upstream unburned value

18

Acknowledgements

This work was supported by the U. S. Department of Energy under Contract DE-AC04-

94AL85000 and by a Memorandum of Understanding between the Office of Munitions (Depart-

ment of Defense) and the Department of Energy. AMT was supported by a President of Russia

Scholarship for Study Abroad.

19

References

1. Baer, M. R., and Nunziato, J. W., Int. J. Multiphase Flow 12:861-889 (1986).

2. Maksimov, E. I., and Merzhanov, A. G., Combust. Eqlos. Sh. Waves 2:25-31 (1966).

3. Merzhanov, A. G., Combust. Flame 13:143-156 (1969).

4. Margolis, S. B., Williams, F. A., and Armstrong, R. C., Combust. Flame 67:249-258 (1987).

5. Li, S. C., Williams, F. A., and Margolis, S. B., Combust. Flame 80:329-349 (1990).

6. Margolis, S. B., and Williams, F. A., Combust. Flame 79:199-213 (1990).

7. Margolis, S. B., and Williams, F. A., J. Propulsion Power 11:759-768 (1995a).

8. Margolis, S. B., and Williams, F. A., Combust. Sei- Technol. 106:41-68 (1995b).

9. Margolis, S. B., and Williams, F. A., Int. J. Multiphase Flow, in press (1996).

10. Prasad, K., Yetter, R. A., and Smooke, M. D., Yale University Report ME-101-95, 1995.

11. Li, S. C., and Williams, F. A., J. Propulsion and Power, in press (1996).

12. Margolis, S. B., and Matkowsky, B. J., SIAM J. Appl. Math. 42:1175-1188 (1982a).

13. Margolis, S. B., and Matkowsky, B. J., Combust. Sci. Technol. 27193-213 (1982b).

14. PelAez, J., and Liiidn, A., SIAM J. Appl. Math. 45:503-522 (1985).

15. Booty, M. L., Holt, J. B., and Matkowsky, B. J., Q. JZ. Mech. Appl. Math. 43:224-249 (1990).

16. Margolis, S. B., Int. J. Eng. Math., submitted (1996).

17. Peldez, J., SIAM J. Appl. Math. 47781-799 (1987)-

20

FIGURE CAPTIONS

Fig. 1. Outer structure of the leftward-propagating deflagration wave. The solid/gas region lies

to the left of [ = 0, and the liquid/gas region to the right. The shaded area denotes the

region [r < [ < [r + H , which, despite the explicit representation afforded by the outer

delta-function formulation, actually lies within the inner reaction zone. The region to

the right of the reaction zone consists of purely gaseous products. Parameter values used

were b = r = Le = 1 = 4 = 1, b = f = 1 = .8, as = .25, QL = 5, Q9 = H = .5, ys = -.2, A ..

T, = 2.

Fig. 2. Inner structure of the leftward-propagating deflagration wave for the case A0 = 4. The

curves were drawn for the parameter regime analyzed in the paper, based on the parameter

values used in Fig. 1 (choosing E = .5, the latter imply that the scaled parameters

ai = CY,/€ = .5, Qb = Q g / c = 1 and i' = (i- 1 ) / ~ = -.4).

Fig. 3. Approximate nondimensional propagation speed U = O(Qs, as)/O(O, 0), corresponding

to the case A0 = 4, as a function of cys for Qg = .1 (solid), .2 (dash), .3 (chain-dash),

.4 (dot) and .5 (chain-dot), where the remaining parameter values were taken to be the

same as those used in the previous figures.

21

0

-10.0 -5.0 5.0 t 0.0

Figure 1

10.0

22

0

4

1

-8.0 -4.0 0.0 77 i .0 8.0

Figure 2

23

U

0 r(

&fa 0

0 0 - 1

0.0 I

0.1 1

0.2 1 1 0.3 Q( 0.4

8

1 0.5

Figure 3

24

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